Non-equilibrium entropy on stationary Markov processes

We study the evolution of probability measures under the action of stationary Markov processes by means of a non-equilibrium entropy defined in terms of a convex function . We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of for a wide class of convex functions, including ₀(t)=t log t. We also prove that this is equivalent to the convergence of all the densities of a finite norm to a uniform density, on the Orlicz spaces related to , which include the L ^p -spaces for p>1. By means of the quadratic function ₂(t)=t ²–1, we relate the non-equlibrium entropies defined by the past -algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.Partially supported by DIB Universidad de Chile, E19468412.     

Entropy Solutions to a Second Order Forward-Backward Parabolic Differential Equation

We prove that the first boundary value problem for a second order forward-backward parabolic differential equation in a bounded domain G _T ^d+1, where d 2, has a unique entropy solution in the sense of F. Otto. Under some natural restrictions on the boundary values this solution is constructed as the limit with respect to a small parameter of a sequence of solutions to Dirichlet problems for an elliptic differential equation. We also show that the entropy solution is stable in the metric of L ₁(G _T ) with respect to perturbations of the boundary values in the metric of L ₁(G _T ).Original Russian Text Copyright © 2005 Kuznetsov I. V.The author was supported by the Russian Foundation for Basic Research (Grant 03-01-00829).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 594–619, May–June, 2005.     

Convergence of flux vector splitting schemes with single entropy inequality for hyperbolic systems of conservation laws

Summary. We prove the convergence of flux vector splitting schemes associated to hyperbolic systems of conservation laws with a single compatible entropy _c. We prove estimate on the L² norm of the gradient of the numerical approximation in the inverse square root of the space increment x. This estimate is related to the notion of (strictly) _c-dissipativity on F⁺, –F^– and Id–(F⁺–F^–), where F⁺, F^– is the flux-decomposition. The second tool of the proof is a kinetic formulation of the flux-splitting scheme with three velocities. Then we get a control for all entropies and apply the compensated compactness theory.Mathematics Subject Classification (2000): 65M12, 35L65, 65M06, 82C40     

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

Spherical multipliers

It is proven in the paper that if functionf(x)L^p(Rⁿ), where 1/p> 1/2 + 1/(n + 1), then the restriction of the Fourier transform f() to the unit sphere S^n–1 lies in L²(S^n–1). As was shown by Fefferman [1], it follows from this that, when > (n –1)/(2(n + 1)), the Riesz-Bochner multiplier acts in LP(Rⁿ) if (n –1–2)/(2n) <1/p < (n + 1 + 2)/(2n).Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 105–112, January, 1978.The author wishes to thank B. S. Mityagin for his attention to this work.     

Fourier series of functions whose Hankel transform is supported on [0, 1]

LetJ denote the Bessel function of order . For >–1, the system x^–/2–1/2J_+2n+1(x^1/2, n=0, 1, 2,..., is orthogonal onL ²((0, ),x dx). In this paper we study the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on [0, 1].Communicated by Mourad Ismail.     

Constraints of the KP hierarchy and the bilinear method

We consider generalizedk-constraints of the KP hierarchy where the Lax operatorL is forced to satisfy L _– ^k =q^–1r. We study the effect of those constraints on the bilinear equations.     

On the Convergence of Stewart's QLP Algorithm for Approximating the SVD

The pivoted QLP decomposition, introduced by Stewart [20], represents the first two steps in an algorithm which approximates the SVD. The matrix A₀ is first factored as A₀=QR, and then the matrix R ^T₁ is factored as R ^T₁=PL ^T, resulting in A=Q₁ LP ^T₀ ^T, with Q and P orthogonal, L lower-triangular, and ₀ and ₁ permutation matrices. Stewart noted that the diagonal elements of L approximate the singular values of A with surprising accuracy. In this paper, we provide mathematical justification for this phenomenon. If there is a gap between _k and _k+1, partition the matrix L into diagonal blocks L ₁₁ and L ₂₂ and off-diagonal block L ₂₁, where L ₁₁ is k-by-k. We show that the convergence of ( _j (L ₁₁)^–1– _j ^–1)/ _j ^–1 for j=1,. . .,k, and of ( _j (L ₂₂)– _k+j )/ _k+j , for j=1,. . .,n–k, are all quadratic in the gap ratio _k+1/ _k . The worst case is therefore at the gap, where the absolute errors L ₁₁ ^–1– _k ^–1 and L ₂₂– _k+1 are thus cubic in _k ^–1 and _k+1, respectively. One order of convergence is due to the rank-revealing pivoting in the first step; then, because of the pivoting in the first step, two more orders are achieved in the second step. Our analysis assumes that ₁=I, that is, that pivoting is done only on the first step. Although our results explain some of the properties of the pivoted QLP decomposition, they hypothesize a gap in the singular values. However, a simple example shows that the decomposition can perform well even in the absence of a gap. Thus there is more to explain, and we hope that our paper encourages others to tackle the problem. The QLP algorithm can be continued beyond the first two steps, and we make some observations concerning the asymptotic convergence. For example, we point out that repeated singular values can accelerate convergence of individual elements. This, in addition to the relative convergence to all of the singular values being quadratic in the gap ratio, further indicates that the QLP decomposition can be powerful even when the ratios between neighboring singular values are close to one.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL _p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW ²:=e ^–2Q , by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW ². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.     

Negative powers of a singular Schrödinger operator and convergence of spectral decompositions

We compare theL ₂( ^N )-norms of negative powers of various Laplace and Schrödinger operators possessing a singular potential whose singularities lie on some manifolds. We write out sufficient conditions for uniform convergence and localization of spectral decompositions of functions from the Liouville class.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 428–436, March, 1996.The author wishes to express deep gratitude to Prof. Sh. A. Alimov for his attention to this work.     

Critical points for spread-out self-avoiding walk,percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in ^d, oriented percolation in ^d×₊, and the contact process in ^d, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y–x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p_c. We investigate the value of p_c when d>6 for percolation and d>4 for the other models, and L1. We prove in a unified way that p_c=1+C(D)+O(L^–2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L^–d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p_c=1+cL^–d+O(L^–d–1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.     

The dynamics of turbo-flow

Zusammenfassung Ausgehend von der Tatsache, dass eine Schaufelreihe äquivalent ist einem Kraft- und damit auch einem Wirbelfeld, wird der Begriff der Potential-Kraft-Strömung erweitert. Diese schliesst den «von Mises-Fall» ein, wenn das Euler-Integral im ganzen Feld konstant ist.Die Struktur des Wirbelfeldes, sowohl des freien als auch des gebundenen, wird analysiert. Das Problem der Turbinenströmung reduziert sich auf die Lösung von zwei partiellen Differentialgleichungen, für die ein vollständiger Satz von Randbedingungen gegeben ist; die Rechnung wird auf bekannte numerische Methoden zurückgeführt.Während die üblichen Rechnungsmethoden nur Eintritts- und Austrittsdiagramme liefern, erhalten wir hier die vollständige Geschwindigkeitsverteilung im ganzen Kraftfeld. Die Schaufelform wird angegeben, und es ist möglich, das Zusammenwirken verschiedener Entwurfsparameter zu untersuchen.

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Nomenclature

Symbol a a constantL ² T ^–2 - B blade surface 0 - C velocity vectorLT ^–1 - C speedLT ^–1 - C _a average axial velocityLT ^–1 - d diameterL - F force per unit massLT ^–2 - f blade shape function 0 - g angular momentum per unit mass (g=C r)L ² T ^–1 - h blade heightL - H Bernoulli's constant (stator)L ² T ^–2 - H ^* Bernoulli's constant (rotor)L ² T ^–2 - k a constantL ² T ^–2 - K a parameterL ^–1 T ^–2 - p pressureML ^–1 T ^–2 - r radius, distance from axisL - R ₀ outer radiusL - R _i hub or inner radiusL - s arc lengthL - t timeT - u a unit vector depending on use - W relative velocityLT ^–1 - load functionL ² T ^–2 - family of surfaces 0 - a parameter 0 - r-component of vorticityT ^–1 - -component of vorticityT ^–1 - angle or direction 0 - a parameter 0 - z-component of vorticityT ^–1 - the number 0 - densityML ^–3 - a functionL ^–1 T ^–1 - direction or angular coordinate in rotors 0 - stream functionL ³ T ^–1 - angular velocityT ^–1 - vorticity vectorT ^–1 - vorticityT ^–1 This work, finished in August 1953, was done while the author was at the Aircraft Gas Turbine Division of the General Electric Company, Cincinnati, Ohio.     

Mean square values of Hecke <Emphasis Type="Italic">L</Emphasis>–series formed with <Emphasis Type="Italic">r</Emphasis>–th order characters

Let L be a number field containing the r–th roots of unity. Starting with the Rankin-Selberg convolution of a metaplectic Eisenstein series on the r–fold cover of GL(2) with itself, we construct a Dirichlet series defined over L whose coefficients involve the r–th order twists of a fixed Hecke L–function. We then observe that a group of functional equations can be naturally associated with this construction. Combining this with the convexity theorem for holomorphic functions of several complex variables, we show that this object, as a function of two complex variables, admits meromorphic continuation to ². As an application, we obtain asymptotic formulae for mean square values of the r–th order twists of an arbitrary Hecke L–function defined over L. Mathematics Subject Classification (1991) 11R42, 11F66, 11F67, 11F70, 11M41, 11R47     

Degrees of logics with Henkin quantifiers in poor vocabularies

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L* is of degree 0. We show that the same holds also for some weaker logics like L(H) and L(E). We show that each logic of the form L^(k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L^(k)(Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H₄) and L(F₂) of Krynicki and M. Mostowski from [KM92].Mathematics Subject Classification (2000): 03C80, 03D35, 03B25Revised version: 28 August 2003     

Expansions for von Mises functionals

Summary Berry-Esseen results and expansions are derived for the distribution function of von Mises functionals of order r under moment conditions and conditions on the smoothness of the limit distribution.The results apply to goodness-of-fit statistics — as well as to the central limit theorem in L ^2p,p2, the rate of convergence being O(n ^–1) for centered balls, provided a fourth moment exists.Research sponsored in part under Office of Naval Research. Contract Number N00014-80-C-0163.     

Convergence of non-ergodic dynamical systems

Summary Let {T _t} be a flow on a probability space (S,L,}) which describes the time evolution of a dynamical system with state space S, and interpret as the initial distribution of the system. Then the distribution of the system at time t is given by T _t ^–1 . Our aim is to study the asymptotic behavior of T _t ^–1 both in general and in the particular cases of random rate and almost periodic systems. The results seem to indicate that convergence or mean convergence is the normal behavior in the non-ergodic case.     

Lifetimes of conditioned diffusions

Summary We investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non-divergence form can be found. The critical value of the parameter is found for each of the following classes of domains:L ^p -domains (p=n–1), uniformly regular twistedL ^p -domains (p=n–1), and twisted Hölder domains (=1/3). A related parabolic boundary Harnack principle is proved.     

Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.